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UBC Theses and Dissertations

Uniform polynomial approximation of even and odd functions on symmetric intervals Dunham, Charles Burton

Abstract

An odd or even continuous function on a symmetric interval [-a,a] can be evaluated in two different ways, each using only one uniform polynomial approximation. It is of practical importance to know which method of evaluation takes fewer arithmetic operations. This is a special case of a more general problem, which is concerned with the optimal subdivision of the interval of evaluation of a function f into sub-intervals, on each of which f has a uniform polynomial approximation. In the first three chapters a method of computing the number of arithmetic operations for evaluation is developed. Expansions in Chebyshev polynomials are studied, with emphasis on the practical problem of computing coefficients, and then it is shown how the expansion in Chebyshev polynomials may be used to obtain truncation error bounds for the uniform polynomial approximation. From these bounds the required degree for the approximation and the required number of multiplications for evaluation may be easily determined. Tables of computed results are given. In Chapter 4 theoretical results are developed from the theory of Lagrange interpolation and these results are in agreement with the computed results obtained previously. In the problem of evaluation of even and odd functions on [-a,a] , use of the uniform polynomial approximation on [-a,a] is advantageous unless the rate of increase of the derivative of f is rapid. In the general case of evaluation of a continuous function, use of approximations on sub-intervals becomes more advantageous the more rapidly the derivatives of f increase.

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